Answer :
Sure! Here's a step-by-step solution to the problem of finding three consecutive odd integers such that eight more than the sum of the first two is equal to eleven less than three times the third. We need to find the third number.
1. Define the three consecutive odd integers:
Let's denote the first odd integer by [tex]\( x \)[/tex].
The next consecutive odd integers will be [tex]\( x + 2 \)[/tex] and [tex]\( x + 4 \)[/tex].
So, we have:
- First integer: [tex]\( x \)[/tex]
- Second integer: [tex]\( x + 2 \)[/tex]
- Third integer: [tex]\( x + 4 \)[/tex]
2. Setup the equation:
According to the problem, eight more than the sum of the first two integers is equal to eleven less than three times the third integer.
This translates to the equation:
[tex]\[
8 + (x + (x + 2)) = 3(x + 4) - 11
\][/tex]
3. Simplify the equation:
- Combine like terms on the left side:
[tex]\[
8 + (x + x + 2) = 8 + (2x + 2)
\][/tex]
[tex]\[
8 + 2x + 2 = 2x + 10
\][/tex]
- Expand the expression on the right side:
[tex]\[
3(x + 4) - 11 = 3x + 12 - 11
\][/tex]
[tex]\[
3x + 1
\][/tex]
- Now we equate the simplified forms on both sides:
[tex]\[
2x + 10 = 3x + 1
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
- Subtract [tex]\( 2x \)[/tex] from both sides to isolate the variable term:
[tex]\[
10 = x + 1
\][/tex]
- Subtract 1 from both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
9 = x
\][/tex]
5. Find the third number:
- The third consecutive odd integer will be [tex]\( x + 4 \)[/tex]:
[tex]\[
x + 4 = 9 + 4 = 13
\][/tex]
Therefore, the third consecutive odd integer is [tex]\( 13 \)[/tex].
The correct answer is: 13
1. Define the three consecutive odd integers:
Let's denote the first odd integer by [tex]\( x \)[/tex].
The next consecutive odd integers will be [tex]\( x + 2 \)[/tex] and [tex]\( x + 4 \)[/tex].
So, we have:
- First integer: [tex]\( x \)[/tex]
- Second integer: [tex]\( x + 2 \)[/tex]
- Third integer: [tex]\( x + 4 \)[/tex]
2. Setup the equation:
According to the problem, eight more than the sum of the first two integers is equal to eleven less than three times the third integer.
This translates to the equation:
[tex]\[
8 + (x + (x + 2)) = 3(x + 4) - 11
\][/tex]
3. Simplify the equation:
- Combine like terms on the left side:
[tex]\[
8 + (x + x + 2) = 8 + (2x + 2)
\][/tex]
[tex]\[
8 + 2x + 2 = 2x + 10
\][/tex]
- Expand the expression on the right side:
[tex]\[
3(x + 4) - 11 = 3x + 12 - 11
\][/tex]
[tex]\[
3x + 1
\][/tex]
- Now we equate the simplified forms on both sides:
[tex]\[
2x + 10 = 3x + 1
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
- Subtract [tex]\( 2x \)[/tex] from both sides to isolate the variable term:
[tex]\[
10 = x + 1
\][/tex]
- Subtract 1 from both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
9 = x
\][/tex]
5. Find the third number:
- The third consecutive odd integer will be [tex]\( x + 4 \)[/tex]:
[tex]\[
x + 4 = 9 + 4 = 13
\][/tex]
Therefore, the third consecutive odd integer is [tex]\( 13 \)[/tex].
The correct answer is: 13
